ScalingStacks

Lemma 7.1.7. The image of a morphism of $1$-dimensional spaces is a $1$-dimensional subspace.

If $Y$ is a $1$-dimensional subspace of $X$, then $Y$ is a $1$-dimensional space and the inclusion map $Y\hookrightarrow X$ is a morphism of $1$-dimensional spaces.

Let $f:X\to X^{\prime}$ be a morphism of $1$-dimensional spaces and $Y^{\prime}$ be a $1$-dimensional subspace of $X^{\prime}$. Let $F$ be the set of connected components of $f^{-1}(Y^{\prime})$ that are points. Then $F$ is finite, $Y=f^{-1}(Y^{\prime})-F$ is a $1$-dimensional subspace of $X$ and $f_{|Y}:Y\to Y^{\prime}$ is a morphism of $1$-dimensional spaces.